Finally, the stage has been set to prove Theorem REF . Let \(n\ge 1\)  be a number. Let \(\Gamma \)  be a graph on \(\mathbb {R}^n\)  without a perfect clique. Let \(\kappa \)  be an inaccessible cardinal and let \(W\)  be the choiceless Solovay model derived from \(\kappa \) ; \(W\)  is a model of ZF+DC [1]}. Let \(P\)  be the coloring poset isolated in Theorem REF . Let \(G\subset P\)  be a filter generic over \(W\) . The model \(W[G]\)  is a \(\sigma \) -closed extension of \(W\)  by Proposition REF  and as such it is a model of ZF+DC. In addition, \(\bigcup G\)  is a total coloring of the graph \(\Gamma \)  with countably many colors by Corollary REF . By Corollary REF  and Theorem REF , in the model \(W[G]\)  every nonmeager subset of \(\mathbb {R}^{n+1}\)  contains point of arbitrary small enough distance. By the Baire category theorem, it must be the case that the chromatic number of \(\Gamma _{n+1}\)  is uncountable in \(W[G]\)  as required. Thus, the model \(W[G]\)  witnesses the conclusion of Theorem REF ; in case that \(\Gamma =\Gamma _n\) , it witnesses the conclusion of Theorem REF .
